Mathematics Commons^™

The Archimedean - January 2013, Department Of Mathematics, Syracuse University

Newsletters from the Department of Mathematics - The Archimedean

No abstract provided.

New Bounds And Computations On Prime-Indexed Primes, Jonathan Bayless, Dominic Klyve, Tomas Oliveira E Silva

Mathematics Faculty Scholarship

In a 2009 article, Barnett and Broughan considered the set of prime-index primes. If the prime numbers are listed in increasing order (2, 3, 5, 7, 11, 13, 17, . . .), then the prime-index primes are those which occur in a prime-numbered position in the list (3, 5, 11, 17, . . .). Barnett and Broughan established a prime-indexed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indexed prime.

We give explicit upper and lower bounds for π²(x), the number of prime-indexed primes up to …

Methods In Visual Mathematics: Reductionism In Researching Mathematical Principles In Art, Lauren N. Colie

AUCTUS: The Journal of Undergraduate Research and Creative Scholarship

People traditionally rely on visual arts as an effective communication tool and medium of self-expression for when words fail to convey abstract concepts. Thera Mjaaland, anthropologist and professional photographer, writes, “Art is capable of negotiating conceptual gaps caused by a dichotomized epistemology” (393). In essence, Mjaaland asserts that art helps relate different modes of thinking by illustrating the abstract and difficult to grasp—privileging the communicative value of an image over that of text. Within this method of communication is a collection of works acknowledged by public consensus to be of an elevated status or value. The art world is deeply …

A New Recursion For Three-Column Combinatorial Macdonald Polynomials, Elizabeth Niese

Mathematics Faculty Research

The Hilbert series of the Garsia–Haiman module M_μ can be described combinatorially as the generating function of certain fillings of the Ferrers diagram of μ where μ is an integer partition of n . Since there are n ! fillings that generate , it is desirable to find recursions to reduce the number of fillings that need to be considered when computing combinatorially. In this paper, we present a combinatorial recursion for the case where μ is an n by 3 rectangle. This allows us to reduce the number of fillings under consideration from (3n)! to (3n)!/(3!ⁿn!).

Rational Map Of Cp^2 With No Invariant Foliation, Scott R. Kaschner, Rodrigo A. Perez, Roland K.W. Roeder

Scholarship and Professional Work - LAS

Conference Poster presented at: Midwest Dynamical Systems Conference, Champaign/Urbana, IL November 1-3, 2013.

Chasing Squares [Pamphlet], Jeremiah Farrell

Scholarship and Professional Work - LAS

Pamphlet accompanying Jeremiah Farrell's mass-produced "Chasing Squares" puzzle.

Nightmares, Jeremiah Farrell, Karen Farrell

Scholarship and Professional Work - LAS

Jeremiah's puzzle "Nightmare", which was exchanged at the 2013 Washington, DC International Puzzle Party. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exchange them. This puzzle was a special puzzle gift that was given to IPP32 exchangers by its designer, Jerry Farrell, in memory of longtime IPP member, Tom Rodgers, Jr.

Flying Saucer, Jeremiah Farrell, Karen Farrell

Scholarship and Professional Work - LAS

Jeremiah's puzzle "Flying Saucer", which was exchanged at the 2013 International Puzzle Party in Washington, DC. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exchange them. This puzzle is also manufactured by Walter Hoppe as "Flying Saucer".

On A Paley-Wiener Theorem For The Zs-Akns Scattering Transform, Ryan D. Walker

Theses and Dissertations--Mathematics

In this thesis, we establish an analog of the Paley-Wiener Theorem for the ZS-AKNS scattering transform on a set of real potentials. We also demonstrate one application of our techniques to the study of an inverse spectral problem for a half-line Miura potential Schroedinger equation.

Young Children Investigating Advanced Mathematical Concepts With Haptic Technologies: Future Design Perspectives, Stephen Hegedus

The Mathematics Enthusiast

In this chapter, we focus on how new technologies can be used with young children to investigate mathematical ideas and concepts that would normally be introduced at a later age. In particular, we focus on haptic technologies that allow learners to touch and feel objects through force feedback in addition to visual images on a screen. The main purpose of this paper is to describe how these technologies can be used to enable young learners to construct meaning about geometric shapes and surfaces as well as attributes of particular mathematical constructions in multiple dimensions (particularly 2D and 3D for purposes …

Smooth Quantile Processes For Right Censored Data, Katsuhiro Uechi

Mathematics Dissertations - Archive

The development of an estimator of a quantile function Q(p) is discussed. The smooth nonparametric estimator Qn(p) of a quantile function Q(p) is defined as the solution to Fn(Qn(p)) = p, where Fn is a smooth Kaplan-Meier estimator of an unknown continuous distribution function F(x). The asymptotic properties of the smooth quantile process, n(Qn(p) - Q(p)) , based on right censored lifetimes are studied. The asymptotic properties of the bootstrap quantile process, n(Q n(p) - Q(p)) are also investigated and shown to have the same limiting distribution as the smooth quantile process. The bootstrap method to approximate the sampling distribution …

The Generalized Two-Component Hunter-Saxton System, Byungsoo Moon

Mathematics Dissertations - Archive

This thesis is concerned with the generalized two-component Hunter-Saxton system. In the periodic setting, we study the wave-breaking phenomenon and global existence for the generalized two-component Hunter-Saxton system. We obtain a brief derivation of the model. We also briefly sketch a standard local well-posednessresult using Kato's semigroup approach. We establish a wave-breaking criterion for solutions and some interesting results of wave-breaking solutions with certain initial profiles. We demonstrate the exact blow-up rate of strong solutions. Finally, we give a sufficient condition for global solutions.

Numerical Integration Of Matrix Riccati Differential Equations With Solution Singularities, Charles K. Garrett

Mathematics Dissertations - Archive

A matrix Riccati differential equation (MRDE) is a quadratic ODE of the form X' = A₂₁ + A₂₂X - XA₁₁ - XA₁₂X ; where X is a function of t with X : R Rnxm and the Aij's are constant or functions of t with matrix sizes to respect the size of X. It is well known that MRDEs may have singularities in their solution even if all the Aij are constant. In this dissertation, several di erent ideas for the meaning of the solution of an MRDE past a solution singularity are analyzed and it is shown how all …

Creating An Interdisciplinary Research Course In Mathematical Ecology, Glenn Ledder, Brigitte Tenhumberg

School of Biological Sciences: Faculty Publications

An integrated interdisciplinary research course in biology and mathematics is useful for recruiting students to interdisciplinary research careers, but there are difficulties involved in creating and implementing it. We describe the genesis, objectives, design policies, and structure of the Research Skills in Theoretical Ecology course at the University of Nebraska–Lincoln and discuss the difficulties that can arise in designing and implementing interdisciplinary courses.

An Interdisciplinary Research Course In Theoretical Ecology For Young Undergraduates, Glenn Ledder, Brigitte Tenhumberg, G. Travis Adams

School of Biological Sciences: Faculty Publications

As part of an interdepartmental effort to attract promising young students to research at the interface between mathematics and biology, we created a course in which groups of recent high school graduates and first-year college students conducted a research project in insect population dynamics. The students set up experiments, collected data, used the data to develop mathematical models, tested their models against further experiments, and prepared their results for dissemination. The course was self-contained in that the lecture portion developed the mathematical, statistical, and biological background needed for the research. A special writing component helped students learn the principles of …

Number Representation And Calculation: An Overview Of Chapter 4 Of Thinking Mathematically, Emily Pawlicki

A with Honors Projects

A presentation providing an overview of numeration systems and using different bases in calculations.

Strongly Gorenstein Flat And Gorenstein Fp-Injective Modules, Chunhua Yang

Turkish Journal of Mathematics

In this paper, we first study the properties of strongly Gorenstein flat (resp. Gorenstein FP-injective) modules which are special Gorenstein projective (resp. Gorenstein injective) modules, and use them to prove that the global strongly Gorenstein flat dimension and the global Gorenstein FP-injective dimension of a ring R are identical when R is n-FC or commutative coherent. Finally, we show that if R is a commutative Noetherian ring, then, for any R-module M, SGfd_RM=Gpd_RM, and hence SGfd_RM< \infty if and only if Gfd_RM

Existence And Multiplicity Of Positive Solutions For A Class Of Nonlinear Elliptic Problems, Asadollah Aghajani, Jamile Shamshiri, Frajollah Mohammadi Yaghoobi

Turkish Journal of Mathematics

We study the existence and multiplicity of nonnegative solutions for the nonlinear elliptic problem, -\Delta u+v(x)u=a(x)u^p+\lambda f(x,u) for x\in\Omega and u=0 on \partial\Omega, where \Omega is a bounded region in R^N, N>2, 1

Finitistic Dimension Conjectures For Representations Of Quivers, Sergio Estrada, Salahatti̇n Özdemi̇r

Turkish Journal of Mathematics

Let R be a ring and Q be a quiver. We prove the first Finitistic Dimension Conjecture to be true for RQ, the path ring of Q over R, provided that R satisfies the conjecture. In fact, we prove that if the little and the big finitistic dimensions of R coincide and equal n

Valuation Of Over-The-Counter (Otc) Derivatives With Collateralization, Leon Guerrero

Electronic Theses and Dissertations

Collateralization in over-the-counter (OTC) derivatives markets has grown rapidly over the past decade, and even faster in the past few years, due to the impact of the recent financial crisis and the particularly important attention to the counterparty credit risk in derivatives contracts. The addition of collateralization to such contracts significantly reduces the counterparty credit risk and allows to offset liabilities in case of default. We study the problem of valuation of OTC derivatives with payoff in a single currency and with single underlying asset for the cases of zero, partial, and perfect collateralization. We assume the derivative is traded …

Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, Tracy Layden

Scripps Senior Theses

This thesis examines the rank of elliptic curves. We first examine the correspondences between projective space and affine space, and use the projective point at infinity to establish the group law on elliptic curves. We prove a section of Mordell’s Theorem to establish that the abelian group of rational points on an elliptic curve is finitely generated. We then use homomorphisms established in our proof to find a formula for the rank, and then provide examples of computations.

Barred Preferential Arrangements, Connor Thomas Ahlbach '13, Jeremy Usatine '14, Nicholas Pippenger

All HMC Faculty Publications and Research

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for r_m‚_ℓ, the number of barred preferential arrangements of ℓ elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r_m,_ℓ as a linear combination of the r_k (= r_0,_ …

Nonlocal Aggregation Models: A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members' intrinsic social interactions with each other and by their interactions with the external environment. Starting from a simple discrete model treating individual organisms as point particles, we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation. To study equilibria and their stability, we use tools from the calculus of variations. In one spatial dimension, and for several choices of social forces, external forces, and domains, we find exact analytical expressions for the equilibria. …

Problem Solving In The Primary School (K-2), Richard Lesh, Lyn English, Chanda Riggs, Serife Sevis

The Mathematics Enthusiast

This article focuses on problem solving activities in a first grade classroom in a typical small community and school in Indiana. But, the teacher and the activities in this class were not at all typical of what goes on in most comparable classrooms; and, the issues that will be addressed are relevant and important for students from kindergarten through college. Can children really solve problems that involve concepts (or skills) that they have not yet been taught? Can children really create important mathematical concepts on their own – without a lot of guidance from teachers? What is the relationship between …

High Jump Analysis, Paige Cooke

Undergraduate Journal of Mathematical Modeling: One + Two

This project presents a mathematical analysis of the high jump, a popular track and field event. The first and second stages of the high jump correspond to the athlete’s run along two distinct trajectories. The third stage is the actual jump. We propose an individual model for each of these stages and show how to combine these models to study the dynamics of the entire high jump.

A Simplified Model Of The Internal Combustion Engine, Christofer Neff

Undergraduate Journal of Mathematical Modeling: One + Two

This project further investigates a model of a simplified internal combustion engine considered by Kranc in 1977. Using Euler’s method for ordinary differential equations, we modeled the interaction between the engine’s flywheel and thermodynamic power cycle. Approximating with sufficiently small time intervals (0.001 seconds over a period of 12 seconds) reproduced Kranc’s results with the engine having an average angular velocity of 72/sec.

Volatilization Of Benzene In A River, Eric Dunlop

Undergraduate Journal of Mathematical Modeling: One + Two

Benzene is a volatile organic compound: when it contaminates a river, some of the substance will evaporate as it flows through. We examine the volumetric flow rate to find how volatilization affects the concentration levels of benzene as the substance flows through several consecutive sections of a river, using a specific example to illustrate the general method.

Influence Of Vectors' Risk-Spreading Strategies And Environmental Stochasticity On The Epidemiology And Evolution Of Vector-Borne Diseaes: The Example Of Chagas' Disease, Christopher Kribs, Perrine Pelosse, Marine Ginoux, Jorge E. Rabinovich, Sebastien Gourbiere, Frederic Menu

Mathematics Faculty Publications - Archive

Insects are known to display strategies that spread the risk of encountering unfavorable conditions, thereby decreasing the extinction probability of genetic lineages in unpredictable environments. To what extent these strategies influence the epidemiology and evolution of vector-borne diseases in stochastic environments is largely unknown. In triatomines, the vectors of the parasite Trypanosoma cruzi, the etiological agent of Chagas’ disease, juvenile development time varies between individuals and such variation most likely decreases the extinction risk of vector populations in stochastic environments. We developed a simplified multi-stage vector-borne SI epidemiological model to investigate how vector riskspreading strategies and environmental stochasticity influence the …

Solving The Optimization Control Problem For Lunar Soft Landing Using Minimization Technique, Lizeth Patricia Ocampo

Mathematics Theses - Archive

Minimizing fuel consumption in lunar missions has been a well studied and documented optimization problem. In this paper two cases of the lunar Lander are studied. The first case is the one dimensional problem where the objective is to make a vertical soft landing using the minimum amount of fuel. The second case has the same objective but an initial tangential velocity greater than zero is given making it a two dimensional problem. The first case is solved using Newton's shooting method, finite difference method (using MATLAB's embedded function bvp4c), and solving it explicitly. For the second case, a minimization …

Point Modules Over Regular Graded Skew Clifford Algebras, Padmini Pillay Veerapen

Mathematics Dissertations - Archive

In this thesis, I consider point modules over regular graded skew Clifford algebras.First, I define a notion of rank (called μ-rank) on noncommutative quadratic forms. To every (commutative) quadratic form is associated a symmetric matrix, and one has the standard notions of rank and determinant function defined on the matrix, and, thus, on the quadratic form. In 2010, in [15], the notion of quadratic form was extended to the noncommutative setting and a one-to-one correspondence was established between these quadratic forms and certain matrices. Using this generalization, I define a notion of rank (called μ-rank) for such noncommutative quadratic forms, …